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Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on fuzzy Aumann's integral theory, the Hermite–Hadamard's type inequalities are introduced for a newly defined class of nonconvex functions, which is known as  U · D preinvex fuzzy number-valued mappings ( U · D preinvex  F · N · V · M s) on coordinates. Some Pachpatte-type inequalities are also established for the product of two  U · D preinvex  F · N · V · M s, and some Hermite–Hadamard–Fejér-type inequalities are also acquired via fuzzy Aumann's integrals. Additionally, several new generalized inequalities are also obtained for the special situations of the parameters. Additionally, some of the interesting remarks are provided to acquire the classical and new exceptional cases that can be considered as applications of the main outcomes. Lastly, a few suggested uses for these inequalities in numerical integration are made. [ABSTRACT FROM AUTHOR]

Computational methods are an important tool for solving the Yang-Baxter equations (in small dimensions), for classifying (unifying) structures and for solving related problems. This paper is an account of some of the latest developments on the Yang-Baxter equation, its set-theoretical version and its applications. We construct new set-theoretical solutions for the Yang-Baxter equation. Unification theories and other results are proposed or proven. [ABSTRACT FROM AUTHOR]

The conditional mean value has applications in regression analysis and in financial mathematics, because they are used in it. We can find papers from recent years that use the conditional mean value in fuzzy cases. As the intuitionstic fuzzy sets are an extension of fuzzy sets, we will try to define a conditional mean value for the intuitionistic fuzzy case. The conditional mean value in crisp intuitionistic fuzzy events was first studied by V. Valenčáková in 2009. She used Gödel connectives. Her approach can only be used for special cases of intuitionistic fuzzy events, therefore, we want to define a conditional mean value for all elements of a family of intuitionistic fuzzy events. In this paper, we define the conditional mean value for intuitionistic fuzzy events using Lukasiewicz connectives. We use a Kolmogorov approach and the notions from a classical probability theory for construction. B. Riečan formulated a conditional intuitionistic fuzzy probability for intuitionistic fuzzy events using an intuitionistic fuzzy state in 2012. In classical cases, there exists a connection between the conditional probability and the conditional mean value, therefore we show a connection between the conditional intuitionistic fuzzy probability induced by the intuitionistic fuzzy state and the conditional intuitionistic fuzzy mean value. [ABSTRACT FROM AUTHOR]

In agent basedmodels, the agents are usually platforms (individual soldiers, tanks, helicopters, etc.), not military units. In the Sandis software, the agents can be platoon size units. As there are about 30 soldiers in a platoon, there is a need for strength distribution in simulations. The contribution of this paper is a conceptual model of the platoon level agent, the needed mathematical models and concepts, and references earlier studies of how simulations have been conducted in a data farming environment with platoon/squad size unit agents with strength distribution. [ABSTRACT FROM AUTHOR]

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. [ABSTRACT FROM AUTHOR]

This article concentrates on exploring the relevance of the postmodernist concept of the event to mathematical philosophy and the foundations of mathematics. In both the scientific and philosophical study of nature, and particularly event ontology, we find that space and dynamism are fundamental. However, whether based on set theory or category theory, modern mathematics faces conceptual and philosophical difficulties when the temporal is intentionally invoked as a key aspect of that intrinsic dynamism so characteristic of mathematical being, physical becoming, process, and thought. We present a multidisciplinary investigation targeting a diverse audience including mathematicians, scientists, and philosophers who are interested in exploring alternative modes of doing mathematics or using mathematics to approach nature. Our aim is to understand both the formal character and the philosophy of time as realized through a radical mode of thinking that goes beyond the spatial in mathematics. In particular, we suggest the need to transcend the purely geometrical view altogether in future foundational research in both mathematics and mathematical philosophy. We reexamine these issues at a fundamental and comprehensive level, where a detailed exposition and critique of both modern set theories and theories of space is outlined, with emphasis on how the philosophy of Idealism has been permeating much of old and new mathematics. Furthermore, toward the end of the article, we explore some possible constructive directions in mathematical ontology by providing new proposals on how to develop a fragment of mathematics for the description of dynamic events. [ABSTRACT FROM AUTHOR]

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f (x) = 0 . In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f (x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence { x n } of approximations to a real root of f (x) , where, for n ≥ k , x n + 1 = x n − f (x n) / p n , k ′ (x n) , p n , k (x) being the polynomial of degree k that interpolates f (x) at x n , x n − 1 , ... , x n − k , the order s k of this method satisfying 1 < s k < 2 . Clearly, when k = 1 , this method reduces to the secant method with s 1 = (1 + 5) / 2 . In addition, s 1 < s 2 < s 3 < ⋯ , such that lim k → ∞ s k = 2 . In this note, we study the application of this method to simple complex roots of a function f (z) . We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples. [ABSTRACT FROM AUTHOR]